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Nested open sets?

  1. Mar 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Give an example of an infinite collection of nested open sets.
    [itex] o_1 \supseteq o_2 \supseteq o_3 \supseteq o_4 ..... [/itex]
    Whose intersection [itex] \bigcap_{n=1}^{ \infty} O_n [/itex] is
    closed and non empty.
    2. Relevant equations
    A set [itex] O \subseteq \mathbb{R} [/itex] is open if for all points, [itex] a \in O [/itex]
    there exists an [itex] \epsilon [/itex] neighborhood [itex] V_{\epsilon}(a) \subseteq O [/itex]
    3. The attempt at a solution
    It seems like if we started with the open interval (0,1) and then took a smaller interval that was nested inside the original interval, and then just kept doing this until we enclosed one point in the interval.
     
  2. jcsd
  3. Mar 18, 2012 #2

    micromass

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    Yes, that's the idea.

    Can you make this rigorous??
     
  4. Mar 18, 2012 #3
    Can I just take the middle one-third of the set.
    so after n operations i will have [itex] \frac{1}{3^n} [/itex]
     
  5. Mar 18, 2012 #4

    jgens

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    I cannot really tell where you are going with that last response. It might be a little easier if you consider intervals of the form [itex](-\frac{1}{n},\frac{1}{n})[/itex].
     
  6. Mar 18, 2012 #5

    micromass

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    What do you mean?? 1/3^n is just a number.

    Just find sets that get smaller and smaller each time.

    start with ]-1,1[, then ]-1/2,1/2[. Then what??
     
  7. Mar 18, 2012 #6
    ok so like jgens said use [itex] ( \frac{-1}{n} , \frac{1}{n}) [/itex]
    And then eventually after n goes to infinity I will have 0 as my enclosed point.
    so if I make an [itex] \epsilon [/itex] radius around 0 i will contain points inside of O the original set. Would the set zero it self be closed be cause if we make
    an [itex] \epsilon [/itex] radius around 0 it wont contain elements that are in the set zero itself.
     
  8. Mar 18, 2012 #7

    HallsofIvy

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    Or, if you really want [itex]1/3^n[/itex], you could use [itex]O_n= (-1/3^n, 1/3^n)[/itex].
     
  9. Mar 18, 2012 #8
    ok, thanks everyone for the help
     
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